New potentials for conformal mechanics

TL;DR

This paper characterizes the most general form of potentials in 1D conformal mechanics with time-independent couplings, exploring their symmetry embeddings and providing examples including Calogero models and gauge theories with AdS backgrounds.

Contribution

It derives the general potential form for 1D conformal systems and classifies their $SL(2,R)$ symmetry embeddings, including conditions for gauge theories to exhibit conformal symmetry.

Findings

General potential form $V=V_0+V_1$ with specific homogeneity properties.

Classification of $SL(2,R)$ embeddings in Diff(R) and Diff(S^1).

Examples include Calogero models and gauge theories with AdS backgrounds.

Abstract

We find under some mild assumptions that the most general potential of 1-dimensional conformal systems with time independent couplings is expressed as $V=V_0+V_1$, where $V_0$ is a homogeneous function with respect to a homothetic motion in configuration space and $V_1$ is determined from an equation with source a homothetic potential. Such systems admit at most an $SL(2,\bR)$ conformal symmetry which, depending on the couplings, is embedded in Diff(R)$ in three different ways. In one case, $SL(2,\bR)$ is also embedded in Diff(S^1). Examples of such models include those with potential $V=\alpha x^2+\beta x^{-2}$ for arbitrary couplings $\alpha$ and $\beta$, the Calogero models with harmonic oscillator couplings and non-linear models with suitable metrics and potentials. In addition, we give the conditions on the couplings for a class of gauge theories to admit a $SL(2,\bR)$ conformal symmetry. We present examples of such systems with general gauge groups and global symmetries that include the isometries of $AdS_2 x S^3$ and $AdS_2 x S^3 x S^3$ which arise as backgrounds in $AdS_2/CFT_1$.